# Page:SahaSpaceTime.djvu/18

APPENDIX

87

upon the second electron e, (likewise moving in any possible manner) at P1 is represented by

${\displaystyle -ee_{1}\left({\dot {t}}_{1}-{\frac {{\dot {x}}_{1}}{c}}\right)\mathrm {F} }$

For the components ${\displaystyle \mathrm {F} _{x},\ \mathrm {F} _{y},\ \mathrm {F} _{z},\ F_{t}}$ of the vector ${\displaystyle F}$ the following three relations hold: —

${\displaystyle c\mathrm {F} _{t}-\mathrm {F} _{x}={\frac {1}{r^{2}}},\ \mathrm {F} _{y}={\frac {\ddot {y}}{c^{2}r}},\ \mathrm {F} _{z}=0}$

and fourthly this vector F is normal to the velocity-vector P1, and through this circumstance alone, its dependence on this last velocity-vector arises.

I£ we compare with this expression the previous formulæ[1] giving the elementary law about the ponderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions; but in three dimensions, they have very complicated projections.

In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics, and modern electrodynamics automatically disappear. I shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses m and m1 describe their world-lines; a moving force-vector is exercised by m upon m1, and the expression is just the same as in the case of the electron, only we have to write +mm1 instead of -ee1. We shall consider only the special case in which the acceleration-vector of m is always zero;

1. K. Schwarzschild. Gött-Nachr. 1903.
H. A. Lorentz, Enzyklopädie der Math. Wissenschaften V. Art 14, p. 199.